Cusp forms for congruence subgroups of $Sp_n(\mathbb{Z})$ and theta functions

In this paper we use theta functions with rational characteristic to construct cusp forms for congruence subgroups $\Gamma _g(p)$ of $Sp(g,\mathbb Z)$.The action of the quotient group $Sp(g,\mathbb Z_p)$ on these forms is conjugate to the linear action of $Sp(g,\mathbb Z_p)$ on $(\mathbb Z_p)^{2g}$. We show that these forms are higher-dimensional analogues of the Fricke functions.